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M ARIO F . T RIOLA

S TATISTICS. E LEMENTARY. Chapter 4 Probability Distributions. M ARIO F . T RIOLA. E IGHTH. E DITION. Chapter 4 Probability Distributions. 4-1 Overview 4-2 Random Variables 4-3 Binomial Probability Distributions

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M ARIO F . T RIOLA

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  1. STATISTICS ELEMENTARY Chapter 4 Probability Distributions MARIO F. TRIOLA EIGHTH EDITION

  2. Chapter 4Probability Distributions 4-1 Overview 4-2 Random Variables 4-3 Binomial Probability Distributions 4-4 Mean, Variance, Standard Deviation for the Binomial Distribution 4-5 The Poisson Distribution

  3. 4-1 Overview This chapter will deal with the construction of probability distributions by combining the methods of Chapter 2 with the those of Chapter 3. Probability Distributions will describe what will probably happen instead of what actually did happen.

  4. Combining Descriptive Statistics Methods and Probabilities to Form a Theoretical Model of Behavior Figure 4-1

  5. 4-2 Random Variables

  6. Definitions • Random Variable a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure • Probability Distribution a graph, table, or formula that gives the probability for each value of the random variable

  7. Table 4-1 Probability DistributionNumber of Girls Among Fourteen Newborn Babies x P(x) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0.000 0.001 0.006 0.022 0.061 0.122 0.183 0.209 0.183 0.122 0.061 0.022 0.006 0.001 0.000

  8. Definitions • Discrete random variable has either a finite number of values or countable number of values, where ‘countable’ refers to the fact that there might be infinitely many values, but they result from a counting process. • Continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale with no gaps or interruptions.

  9. Probability Histogram Figure 4-3

  10. Requirements for Probability Distribution  P(x) = 1 where x assumes all possible values

  11. Requirements for Probability Distribution  P(x) = 1 where x assumes all possible values 0 P(x) 1 for every value of x

  12. Mean, Variance and Standard Deviation of a Probability Distribution Formula 4-1 µ = [x•P(x)] Formula 4-2 2= [(x - µ)2 • P(x)] Formula 4-3 2=[x2 • P(x)] - µ2(shortcut)

  13. Mean, Variance and Standard Deviation of a Probability Distribution Formula 4-1 µ = [x•P(x)] Formula 4-2 2= [(x - µ)2 • P(x)] Formula 4-3 2=[x2 • P(x)] - µ2(shortcut) Formula 4-4 =[x 2 • P(x)] - µ2

  14. Mean, Variance and Standard Deviation of a Probability Distribution Formula 4-1 µ = [x•P(x)] Formula 4-2 2= [(x - µ)2 • P(x)] Formula 4-3 2=[x2 • P(x)] - µ2(shortcut) Formula 4-4 =[x 2 • P(x)] - µ2

  15. Roundoff Rule for µ, 2, and Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of xare integers, round µ, 2, and  to one decimal place.

  16. Definition Expected Value The average value of outcomes E =  [x • P(x)]

  17. E =  [x • P(x)] Event Win Lose

  18. E =  [x • P(x)] Event Win Lose x $499 - $1

  19. E =  [x • P(x)] Event Win Lose x $499 - $1 P(x) 0.001 0.999

  20. E =  [x • P(x)] Event Win Lose x $499 - $1 P(x) 0.001 0.999 x • P(x) 0.499 - 0.999

  21. E =  [x • P(x)] Event Win Lose x $499 - $1 P(x) 0.001 0.999 x • P(x) 0.499 - 0.999 E = -$.50

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